Sample Paper for MA / M.Sc Mathematics of Complex AnalysisItsMyAcademy.com

# Sample Paper for MA / M.Sc Mathematics of Complex Analysis !!

Sample Paper for MA/M.Sc Mathematics of Complex Analysis

TERM END EXAMINATIONS

M. A. /M. Sc. – MATHEMATICS, YEAR – 1

COMPLEX ANALYSIS

#### Duration -3 Hours                                                   Max Marks: 60

Note:   1. Attempt any FIVE questions.

2. All questions carry equal marks.

1.         (a) Show that function : f(z) = (z ¹ 0), f(0) = 0 is not analytic at z = 0, although C.R. equations satisfied at that point.

(b) Derive the formula for the radius of convergence of the power series an zn f find the same for the series.

2.         (a Using Cauchy’s integral formula, prove that  where C is the circle |z| = 3

(b) State & prove weiestrass theorem concerning the behaviour of an analytic function near an isolatial essential singularity.

3.         (a) State & prove Argument principle

(b) Prove that every polynomial of degree n has no zeros & determine the number of roots of the equation

z8 – 4z5 + z2 – 1 = 0 that lies inside the circle |z| = 1

4.         (a) State & prove minimum modulus principle.

(b) Show that cot z =

5.         (a) State & prove Taylor’s Theorem.

(b) Expand f(z) =  in Lament’s series valid for the regions

| < |z+1| < 2

6.         (a) State & prove Fundamental theorem of Algebra with the help of Morera theorem.

(b) Expand log |1+z| is Taylor’s series about z = 0 & determine the region of convergence for the resulting series.

7.         (a) There cannot be two different direct analytic continuations of a function

(b) If f(z) is memorphic inside a closed contour c & has no zero on c then

= N-P

N is the number of zeros & P the number of poles inside C.

8.         (a) State & prove Rouche’s theorem,

(b) State & prove Mittag-Leffler Theorem.